Stochastic dynamics driven by combined Lévy–Gaussian noise: fractional Fokker–Planck–Kolmogorov equation and solution

Abstract

Starting with a stochastic differential equation driven by combined Gaussian and Lévy noise terms we determine the associated fractional Fokker–Planck–Kolmogorov equation (FFPKE). For constant and power-law forms of an external potential we study the interplay of the two noise forms. Particular emphasis is paid on the discussion of sub- and superharmonic external potentials. We derive the probability density function solving the FFPKE and confirm the obtained shapes by numerical simulations. Particular emphasis is also paid to the stationary probability density function in the confining potentials and the question, to which extent the additional Gaussian noise effects changes on the probability density function compared to the pure Lévy noise case.